Here is my advice on how you might try setting up your camera, to see if it helps achieve something closer to the look you are striving for:

Try center-weighted metering rather than multi or wide metering. Use Program Auto (P) rather than full auto. Set EV to 0, or possibly even -.3. Go into picture settings, and choose the Vivid mode, then adjust the sharpness to +1...you can probably leave saturation and contrast at 0...though try adjusting either if it still doesn't pop enough for you. Set your DRO/HDR option to DRO Auto. You can probably leave WB on Auto, but occasionally bring it up and try some of the presets if the scene doesn't look warm enough for your preference - that way, you can add some warmth if you feel it needs it.

You are used to a P&S-style JPG - one that's been tuned very warm, contrasty, heavily saturated, and extremely sharpened. There's nothing wrong with liking what you like, but most larger-sensored cameras tend towards much more subdued and 'true' color and contrast, leaving the ability for the photographer to 'tweak' as he likes. It takes a bit of an adjustment to go from a P&S to a DSLR/mirrorless style, because of this intrinsic difference in how the default processing tends to be.

Just to try to figure out what types of look you prefer, do any of these NEX photos have the warmth or saturation or depth of field or contrast style you prefer?

YES, you are understanding me this is what I like, all of them are lovely and I especially love the bird. The top shot is great bc it shows the icy mountain contrast in the back too.

If you do settle on some jpg settings that give you results that are to your liking please let us know as I quite like the look you are describing and would be interested in trying them out myself.

I always photograph in RAW + JPG though - RAW gives you much greater latitude in post-processing. Do you use Adobe Lightroom? With RAW + Lightroom you should be able to achieve any look you want. I create a preset that applies the lens profile, my desired noise reduction and sharpening, and 'auto tone' and tweak from there if I don't like what auto tone does.